a simulation of the law of flowering plants: a flower first blooms in the spring after cumulative temperatures reach a threshold
Two assumptions
Let \(X_i > 0\) denote the force exerted on the plant on day \(i\).
Assume
\(X_i\) have common mean \(\mu\) (the ambient temperature).
\(X_i\) are independent with common variance \(\sigma^2\).
Both assumptions can be relaxed.
e.g., 1. can be replaced with the assumption that \(X_i\) have mean \(\mu_i\), and \(\sum \mu_i\) does not grow too quickly (regularly varying with index 1).
CLT for stopped random walks
Let \(S_{a}^i = \sum_{j=a}^i X_j\) denote the cumulative force from day \(a\) to \(i\)
Let \(n_{\gamma}\) denote the bloom date.
Assume the plant blooms when \(S_a^i\) first passes \(\gamma\),
\[n_{\gamma} = \min \{ i : S_a^i \geq \gamma \}\] When \(\gamma\) is large,
(figure_marsham <- marsham %>%ggplot() +aes(spring.temp, response.time) +geom_point(alpha = .5) +labs(x ="Cumulative Daily Temperature (°C, 2/1 to 4/30)",y ="Time until Budburst (number of days after January 1)") +theme_bw() +ylim(70, 180))
data from Marsham (1746-1958)
Fit using lm/glm
expand R code
figure_marsham +geom_smooth(aes(weight = spring.temp^3), method ="lm", formula = y ~I(1/x), color ="red") +geom_smooth(method ="glm", formula = y ~I(1/x),method.args =list(family =inverse.gaussian(link ="identity")))
In experimental setting, day forcing begins known by design.
For observational, day forcing begins not known.
Cumulative temperature from 2/1 to 4/30 is a proxy for total force. Proxy is accurate when \(\gamma\) large. Why?
Recall \(n_{\gamma} \ \dot \sim \ \text{Normal} \left ( \gamma / \mu + a, \, \gamma \sigma^2 / \mu^3 \right)\) when \(\gamma\) is large.
When \(\gamma\) is large, \(n_{\gamma}\) is large, and \(n\mu \approx \sum_{i=a}^n \mu_i \approx \sum_{i=b}^n \mu_i\) for any \(a,b << n\). i.e., First few \(\mu_i\) don’t matter.
Argument fails when \(\gamma\) is not large.
Non-asymptotic model
Assume instead \(X_i > 0\) is normal with mean \(\mu_i\) and variance \(\sigma^2\).
Note that \(\mathbb{P} \left (n_{\gamma} \leq m \right ) = \mathbb{P} \left (S_a^m > \gamma \right) = \Phi \left( \frac{\sum_{i=a}^m \mu_i - \gamma}{ \sqrt{m - a} \, \sigma} \right)\) so that the likelihood contribution of each observation is \[\mathcal{L} \left (\gamma, \sigma, a ; \, n_{\gamma}, \{\mu_i\}\right ) = \Phi \left( \frac{\sum_{i=a}^{n_{\gamma}} \mu_i - \gamma}{\sqrt{n_{\gamma}-a} \, \sigma} \right ) - \Phi \left( \frac{\sum_{i=a}^{n_{\gamma}-1} \mu_i - \gamma}{\sqrt{n_{\gamma}-1-a} \, \sigma} \right )\]
tibble(data =c(rep("Charrier", 2), rep("Marsham", 2)), model =rep(c("asymptotic", "non-asymptotic"), 2),gamma =c(mean(extract(fit1, "gamma")[[1]]),mean(extract(fit3, "gamma")[[1]]),mean(extract(fit2, "gamma")[[1]]) /90,mean(extract(fit4, "gamma")[[1]])),lower =c(quantile(extract(fit1, "gamma")[[1]], p = .025),quantile(extract(fit3, "gamma")[[1]], p = .025),quantile(extract(fit2, "gamma")[[1]], p = .025) /90,quantile(extract(fit4, "gamma")[[1]], p = .025)),upper =c(quantile(extract(fit1, "gamma")[[1]], p = .975),quantile(extract(fit3, "gamma")[[1]], p = .975),quantile(extract(fit2, "gamma")[[1]], p = .975) /90,quantile(extract(fit4, "gamma")[[1]], p = .975))) %>%kable(align ="c", digits =0,col.names =c("data", "model", "$\\widehat{\\gamma}$","$\\widehat{\\gamma}_{\\text{lower}}$","$\\widehat{\\gamma}_{\\text{upper}}$"))
data
model
\(\widehat{\gamma}\)
\(\widehat{\gamma}_{\text{lower}}\)
\(\widehat{\gamma}_{\text{upper}}\)
Charrier
asymptotic
769
467
1069
Charrier
non-asymptotic
992
806
1186
Marsham
asymptotic
247
213
282
Marsham
non-asymptotic
383
343
417
Summary
We justified modeling biological process as stopped random walks.
We reviewed the CLT for stopped random walks.
We applied the CLT to experimental and observational data.
Found CLT approximation compared well to non-asymptotic model.
The model can be complicated to allow for more covariates or additional variation.
References
Auerbach, Jonathan. (2023). A demonstration of the law of the flowering plants. Real World Data Science. https://realworlddatascience.net/ideas/tutorials/posts/2023/04/13/flowers.html
Charrier, G., Bonhomme, M., Lacointe, A., & Améglio, T. (2011). Are budburst dates, dormancy and cold acclimation in walnut trees (Juglans regia L.) under mainly genotypic or environmental control?. International journal of biometeorology, 55(6), 763-774. https://pubmed.ncbi.nlm.nih.gov/21805380/
Marsham, R. (1789). XIII. Indications of spring, observed by Robert Marsham, Esquire, FRS of Stratton in Norfolk. Latitude 52° 45’. Philosophical Transactions of the Royal Society of London, (79), 154-156.